1
2
3
47
6
Journal of Integer Sequences, Vol. 13 (2010),
Article 10.3.8
23 11
Six Little Squares and How Their
Numbers Grow
Matthias Beck1
Department of Mathematics
San Francisco State University
1600 Holloway Avenue
San Francisco, CA 94132
USA
beck@math.sfsu.edu
Thomas Zaslavsky2
Department of Mathematical Sciences
Binghamton University
Binghamton, NY 13902-6000
USA
zaslav@math.binghamton.edu
Abstract
We count the 3 × 3 magic, semimagic, and magilatin squares, as functions either of
the magic sum or of an upper bound on the entries in the square. Our results on magic
and semimagic squares differ from previous ones, in that we require the entries in the
square to be distinct from each other and we derive our results not by ad hoc reasoning,
but from the general geometric and algebraic method of our paper “An enumerative
geometry for magic and magilatin labellings”. Here we illustrate that method with a
detailed analysis of 3 × 3 squares.
1
2
Research partially supported by National Science Foundation grant DMS-0810105.
Research partially supported by National Science Foundation grant DMS-0070729 and by the SGPNR.
1
1
Introduction
“Today, the study of magic squares is not regarded as a subject of mathematics, but many
earlier mathematicians in China and Japan studied it.” These words from Shigeru’s history of
old Japanese mathematics [14, p. 435] are no longer completely true. While the construction
of magic squares remains for the most part recreational, their counting has become part of
the mainstream of enumerative combinatorics, as an example of quasipolynomial counting
formulas and as an application of Ehrhart’s theory of lattice points in polytopes. There are
several classical [12, 11, 16] and recent [1, 2] mathematical works on counting something like
magic squares, but without the requirement that the entries be distinct, and often omitting
the diagonals. In previous articles [5, 6] we established the groundwork for an enumerative
theory of magic squares with distinct entries. Here we apply those geometrical and algebraic
methods to solve the problem of counting three kinds of magical 3 × 3 squares.
Each square x = (xjk )3×3 has positive integral entries that satisfy certain line-sum equations and distinctness conditions. In a weakly semimagic square every row and column sum
is the same (their common value is called the magic sum); in a weakly magic square each of
the two diagonals also adds up to the magic sum. Such squares have been studied before (see,
e.g., Beck et al. [2] and Stanley [17]); the difference here is that we count strongly magic or
semimagic squares, where all entries of the square are distinct. (Since strongly magic squares
are closest to what are classically known as “magic squares”—see the introduction to our
general magic article [6]—we call strong squares simply “magic” or “semimagic” without
qualification.) The third type we count is a magilatin square; this is a weakly semimagic
square with the restriction that the entries be distinct within a row or column. The numbers
of standard magic squares (with entries 1, 2, . . . , n2 ) and latin squares (in which each row or
column has entries 1, 2, . . . , n) are special evaluations of our counting functions.
We count the squares in two ways: by magic sum (an affine count), and by an upper
bound on the numbers in the square (a cubic count). Letting N (t) denote the number of
squares in terms of a parameter t which is either the magic sum or a strict upper bound on
the entries, we know by our previous work [6] that N is a quasipolynomial, that is, there are
a positive integer p and polynomials N0 , N1 , . . . , Np−1 so that
N (t) = Nt
(mod p) (t)
.
The minimal such p is the period of N ; the polynomials N0 , N1 , . . . , Np−1 are the constituents
of N , and N0 is the principal constituent. Here we find an
of constituents
Pexplicit list
t
and also the explicit rational generating function N(x) =
t>0 N (t)x (from which the
quasipolynomial is easily extracted).
Each magic and semimagic square also has an order type, which is the arrangement of
the cells in order of increasing value of their entries. The order type is a linear ordering of
the cells because all entries are distinct in these squares. There are 9! = 362880 possible
linear orderings but only a handful are order types of squares. Our approach finds the actual
number of order types for each kind of square; it is the absolute value of the constant term
of the principal constituent, that is, |N0 (0)|. (See Theorems 3.4 and 3.14 and Examples
3.11, 3.12, and 3.21 in our paper on magic labellings [6].) Obviously, this number will be the
same for cubic and affine counts of the same kind of square. (There is also an order type for
2
magilatin squares, which is a linear ordering only within each row and column. As it is not
a simple permutation of the cells, we shall not discuss it any further.)
One of our purposes is to illustrate the technique of our general treatment [6]. Another is
to provide data for the further study of magic squares and their relatives; to this end we list
the exact numbers of each type for small values of the parameter and also the numbers of
symmetry types, reduced squares, and reduced symmetry types of each type (and we refer to
the On-Line Encyclopedia of Integer Sequences (OEIS) [15] for the first 10,000 values of each
counting sequence). A square is reduced by subtracting the smallest entry from all entries;
thus, the smallest entry in a reduced square is 0. A square is normalized by being put into
a form that is unique in each symmetry class. Clearly, the number of normalized squares,
i.e., of equivalence classes under symmetry, is fundamental; and the number of reduced,
normalized squares is more fundamental yet.
There are other ways to find exact formulas. Xin [18] tackles 3×3 magic squares, counted
by by magic sum, using MacMahon’s partition calculus. He gets a generating function that
agrees with ours (thereby confirming both). Stanley’s idea of Möbius inversion over the
partition lattice [17, Exercise 4.10] is similar to ours in spirit, but it is less flexible and
requires more computation. Beck and van Herick [3] have counted 4 × 4 magic squares using
the same basic geometrical setup as ours but with a more direct counting method.
Our paper is organized as follows. Section 2 gives an outline of our theoretical and
computational setup, as well as some comments on checks and feasibility. In Section 3 we
give a detailed analysis of our computations for counting magic 3 × 3 squares. Sections 4
and 5 contain the setup and the results of similar computations for 3 × 3 semimagic and
magilatin squares. We conclude in Section 6 with some questions and conjectures.
We hope that these results, and still more the method, will interest both magic squares
enthusiasts and mathematicians.
2
2.1
The technique
General method
The means by which we solve the specific examples of 3 × 3 magic, semimagic, and magilatin
squares is inside-out Ehrhart theory [5]. That means counting the number of 1/t-fractional
points in the interior of a convex polytope P that do not lie in any of a certain set H of
hyperplanes. The number of such points is a quasipolynomial function EP◦ ◦ ,H(t), the open
Ehrhart quasipolynomial of the open inside-out polytope (P ◦ , H). The exact polytope and
hyperplanes depend on which of the six problems it is, but we can describe the general
picture. First, there are the equations of magic; they determine a subspace s of all 3 × 3
real matrices which we like to call the magic subspace—though mostly we work in a smaller
overall space Rd that results from various reductions. Then there is the polytope P of
2
constraints, which is the intersection
with s of either a hypercube [0, 1]3 or a standard
P
2
simplex {x ∈ R3 : x ≥ 0,
xjk = 1}: the former when we impose an upper bound on the
magic square entries and the latter when we predetermine the magic sum. The parameter t
is the strict upper bound in the former case (which we call cubical due to the shape of P ), the
magic sum in the latter (which we call affine as P lies in a proper affine subspace). Finally
3
there are the strong magical exclusions, the hyperplanes that must be avoided in order to
ensure the entries are distinct—or in the magilatin examples, as distinct as they ought to be.
These all have the form xij = xkl . The combination of P and the excluded hyperplanes forms
the vertices of (P, H), which are all the points of intersection of facets of P and hyperplanes
in H that lie in or on the boundary of P . Thus, we count as a vertex every vertex of P itself,
each point that is the intersection of some facets and some hyperplanes in H, and any point
that is the intersection of some hyperplanes and belongs to P , but not intersection points
that are outside P . (Points of each kind do occur in our examples.) The denominator of
(P, H) is the least common denominator of all the coordinates of all the vertices of (P, H).
The period of EP◦ ◦ ,H(t) divides the denominator; this gives us a known bound on it.
This geometry might best be explained with an example. Let us consider magic 3 × 3
squares,


x11 x12 x13
2
 x21 x22 x23  ∈ Z3>0
.
x31 x32 x33
The magic subspace is





 x11 x12 x13
2
s =  x21 x22 x23  ∈ R3 :


 x31 x32 x33
x11 + x12 + x13 = x21 + x22 + x23
= x31 + x32 + x33 = x11 + x21 + x31
= x12 + x22 + x32 = x13 + x23 + x33
= x11 + x22 + x33 = x13 + x22 + x31
The hyperplane arrangement H that captures the distinctness of the entries is




.



H = {(x11 = x12 ) ∩ s, (x11 = x21 ) ∩ s, . . . , (x32 = x33 ) ∩ s} .
Finally, there are two polytopes associated to magic 3 × 3 squares, depending on whether
we count them by an upper bound on the entries:
2
Pc = s ∩ [0, 1]3 ,
or by magic sum:




 x11 x12 x13
2
Pa = s ∩  x21 x22 x23  ∈ R3≥0 : x11 + x12 + x13 = 1 .


x31 x32 x33
Our cubical counting function computes the number of magic squares all of whose entries
satisfy 0 < xij < t, in terms of an integral parameter t. These squares are the lattice points
in
[ 32
H ∩ 1t Z .
Pc◦ \
Our second, affine, counting function computes the number of magic squares with positive
entries and magic sum t. These squares are the lattice points in
[ 32
Pa◦ \
H ∩ 1t Z .
4
In general, the number of squares we want to count, N (t), is the Ehrhart quasipolynomial EP◦ ◦ ,H(t) of an open inside-out polytope (P ◦ , H). We obtain the necessary Ehrhart
quasipolynomials by means of the computer program LattE [9]. It computes the closed
Ehrhart generating function
EP (x) := 1 +
∞
X
EP (t)xt
of the values
EP (t) := # P ∩
t=1
d
1
Z
t
.
Counting only interior points gives the open Ehrhart quasipolynomial EP ◦ (t) and its generating function
∞
X
EP ◦ (x) :=
EP ◦ (t)xt .
t=1
P
◦
t
Since we want the open inside-out Ehrhart generating function E◦P ◦ ,H(x) = ∞
t=1 EP ◦ ,H(t)x ,
we need several transformations. One is Ehrhart reciprocity [10], which is the following
identity of rational generating functions:
EP ◦ (t) = (−1)1+dim P EP (x−1 ) .
(1)
The inside-out version [5, Equation (4.6)] is
E◦P ◦ ,H(x) = (−1)1+dim P EP,H(x−1 ) .
(2)
We need to express the inside-out generating functions in terms of ordinary Ehrhart generating functions. To do that we take the intersection poset
T
L(P ◦ , H) := P ◦ ∩ S : S ⊆ H \ ∅ ,
which is ordered by reverse inclusion. Note that L(P ◦ , H) and L(P, H), defined similarly
but with P instead of P ◦ , are isomorphic posets because H is transverse to P ; specifically,
L(P, H) = {ū : u ∈ L(P ◦ , H)}, where ū is the (topological) closure of u. Now we have the
Möbius inversion formulas [5, Equations (4.7) and (4.8)]
X
E◦P ◦ ,H(x) =
µ(0̂, u) Eu (x)
(3)
u∈L(P ◦ ,H)
and
EP,H(x) =
X
|µ(0̂, u)| Eu (x)
(4)
u∈L(P,H)
(since H is transverse to P ; see our general paper [6]). Here µ is the Möbius function of
L(P ◦ , H) [17].
Thus we begin by getting all the cross-sectional generating functions Eu (x) from LattE.
Then we either sum them by (4) and apply inside-out reciprocity (2), or apply ordinary reciprocity (1) first and then sum by (3). (We did whichever of these seemed more convenient.)
In the semimagic and magilatin counts we need a third step because the generating functions we computed pertain to a reduced problem; those of the original problem are obtained
through multiplication by another generating function.
5
Once we have the generating function we extract the quasipolynomial, essentially by the
binomial series. If an Ehrhart quasipolynomial q of a rational convex polytope has period
p and degree d, then its generating function q can be written as a rational function of the
form
X
ap(d+1) xp(d+1) + ap(d+1)−1 xp(d+1)−1 + · · · + a1 x
q(x) :=
q(t) xt =
(5)
p )d+1
(1
−
x
t≥1
for some nonnegative integers a1 , a2 , . . . , ap(d+1) . Grouping the terms in the numerator of (5)
according to the residue class of the degree modulo p and expanding the denominator, we
get
" d
Pp Pd
#
p
pj+r
X
X X
d+k−j
j=0 apj+r x
r=1
apj+r
=
xpk+r .
q(x) =
d
(1 − xp )d+1
r=1 k≥0 j=0
Hence the rth constituent of the quasipolynomial q is
qr (t) =
d
X
j=0
2.2
apj+r
d+
t−r
p
d
−j
for r = 1, . . . , p.
How we apply the method
The initial step is always to reduce the size of the problem by applying symmetry. Each
problem has a normal form under symmetry, which is a strong square. The number of all
magic or semimagic squares is a constant multiple of the number of symmetry types, because
every such square has the same symmetry group. For magilatin squares, there are several
symmetry types with symmetry groups of different sizes, so each type must be counted
separately.
Semimagic and magilatin squares also have an interesting reduced form, in which the
values are shifted by a constant so that the smallest cell contains 0; and a reduced normal
form; the latter two are not strong but are aids to computation. Reduced squares are counted
either by magic sum (the “affine” counting rule) or by the largest cell value (the “cubic”
count). All reduced normal semimagic squares correspond to the same number of unreduced
squares, while the different symmetry types of magilatin square give reduced normal squares
whose corresponding number of unreduced squares depends on the symmetry type.
The total number of squares, N (t), and
P the number of reduced squares, R(t), are connected by a convolution identity N (t) = s f (t − s)R(s), where f is a periodic constant (by
which we mean a quasipolynomial of degree 0; we say constant term for the degree-0 term of
a quasipolynomial, even though the “constant term” may
or a linear polyP vary periodically)
t
nomial. P
Writing for the generating functions N(x) := t>0 N (t)x and similarly R(x), and
f (x) := t≥0 f (t)xt , we have N(x) = f (x)R(x). It follows from the form of the denominator
in Equation (5) that the period of N divides the product of the periods of f and R. The
reduced number R(t) is, in the semimagic case, a constant multiple of the number n(t) of
reduced, normal semimagic squares; in the magilatin case it is a sum of different multiples,
depending on a symmetry group, of the number of reduced, normal magilatin squares of each
6
different type T . Each n(t) is the open Ehrhart quasipolynomial EQ◦ ◦ ,I(t) of an inside-out
polytope (Q, I) which is smaller
than the original polytope P .
P
We compute n(x) := t>0 n(t)xt from the Ehrhart generating functions Eu (x) of the
nonvoid sections u of Q◦ by flats of I through the following procedure:
1. We calculate the flats and sections by hand.
2. We feed each u into the computer program LattE [9], which returns the closed generating function Eū (x), whose constant term equals 1 because u is nonvoid and convex.
3. With semimagic squares, by Equations (1)–(4) we have the Möbius-inversion formulas
X
n(x) = E◦Q◦ ,I(x) =
µL(0̂, u)Eu (x)
u∈L
= (−1)1+dim Q
X
|µL(0̂, u)| Eū (x−1 ).
(6)
u∈L
◦
where L := L(Q , I), the intersection poset of (Q◦ , I).
The procedure for magilatin squares is similar but taking account of the several types.
2.3
Checks
We check our results in a variety of ways.
The degree is the dimension of the polytope, or the number of independent variables in
the magic-sum equations.
The leading coefficient is the volume of the polytope. (The volume is normalized so that
a fundamental domain in the affine space spanned by the polytope has unit volume.) This
check is also not difficult. The volume is easy to find by hand in the magic examples. In affine
semimagic the polytope is the Birkhoff polytope B3 , whose volume is well known (Section
4.2). The cubical semimagic volume is not well known but it was easy to find (Section 4.1.1).
The magilatin polytopes are the same as the semimagic ones.
The firmest verification is to compare the results of the generating function approach
with those of direct enumeration. If we count the squares individually for t ≤ t1 where
t1 ≥ pd, only the correct quasipolynomial can agree with the counts (given that we know
the degree d and period p from the geometry). Though t1 = pd is too large to reach in
some of the examples, still we gain considerable confidence if even a smaller value of t1 yields
numbers that agree with those derived from the quasipolynomial or generating function. We
performed this check in each case.
2.4
Feasibility
Based on our solutions of the six 3 × 3 examples we believe our counting method is practical.
The calculations are simple and readily verified. Linear algebra tells us the degree, geometry
tells us the period; we obtain the generating function using the Ehrhart package LattE [9]
and then apply reciprocity (Equation (1) or (2)) and Möbius inversion (Equation (3) or (4)),
and extract the constituents, all with Maple. The programming is not too difficult.
7
In the magic square problems we found the denominator by calculating the vertices of
the inside-out polytope. Then we took two different routes. In one we applied LattE and
Equation (3). In the other we calculated N (t) for small values of t by generating all magic
squares, taking enough values of t that we could fit the quasipolynomial constituents to the
data. This was easy to program accurately and quick to compute, and it gave the same
answer. The programs can be found at our “Six Little Squares” Web site [7].
In principle the semimagic and magilatin problems can be solved in the same two ways.
The geometrical method with Möbius inversion gave complete answers in a few minutes
of computer time after a simple hand analysis of the geometry (see Section 4). Direct
enumeration on the computer proved unwieldy (at best), especially in the affine case, where
the period is largest. A straightforward computer count of semimagic squares by magic
sum (performed in Maple—admittedly not the language of choice—on a personal computer)
seemed destined to take a million years. Switching to a count of reduced normal squares, the
calculation threatened to take only a thousand years. These programs are at our “Six Little
Squares” Web site [7], as is a complicated “supernormalized formula” that greatly speeds up
affine semimagic counting (see Section 4.2.7).
2.5
Notation
We use a lot of notation. To keep track of it we try to be reasonably systematic.
M, m refer to magic squares (Section 3).
S, s and subscript s refer to semimagic squares (Section 4).
L, l and subscript ml refer to magilatin squares (Section 5).
R, r refer to reduced squares (the minimum entry is 0), while M, S, L, et al. refer to
ordinary squares (all positive entries).
c refers to “cubic” counting, by an upper bound on the entries.
a refers to “affine” counting, by a specified magic sum.
X (capital) refers to all squares of that type.
x (minuscule) refers to symmetry types of squares, or equivalently normalized squares.
8
3
Magic squares of order 3
The standard form of a magic square of order 3 is well known; it is
α+γ
−α − β + γ
β+γ
−α + β + γ
γ
α−β+γ
−β + γ
α+β+γ
−α + γ
(7)
where the magic sum is s = 3γ. Taking account of the 8-fold symmetry, under which we may
assume the largest corner value is α + γ and the next largest is β + γ, and the distinctness
of the values, we have α > β > 0 and α 6= 2β. One must also have γ > α + β to ensure
positivity.
In this pair of examples, the dimension of the problem is small enough that there is no
advantage in working with the reduced normal form (where γ = 0).
3.1
Magic squares: Cubical count (by upper bound)
Here we count by a strict upper bound t on the permitted values; since the largest entry is
α + β + γ, the bound is α + β + γ < t. The number of squaresSwith upper bound t is Mc (t).
We think of each magic square as a t−1 -lattice point in Pc◦ \ Hc , the (relative) interior of
the inside-out polytope
Pc := {(x, y, z) : 0 ≤ y ≤ x, x + y ≤ z, x + y + z ≤ 1},
Hc := {h} where h : x = 2y,
but multiplied by t to make the entries integers. Here we use normalized coordinates x = α/t,
y = β/t, and z = γ/t. The semilattice of flats is L(Pc◦ , Hc ) = {Pc◦ , h ∩ Pc◦ } with Pc◦ < h ∩ Pc◦ .
The vertices are
O = (0, 0, 0),
C = (0, 0, 1),
D = ( 21 , 0, 12 ),
E = ( 13 , 16 , 21 ),
F = ( 14 , 41 , 21 ),
of which O, C, D, F are the vertices of Pc and O, C, E are those of h ∩ Pc . (Both these
polytopes are simplices.) From Equation (3),
Mc (x) = 8E◦Pc◦ ,Hc (x) = 8 EPc◦ (x) − Eh∩Pc◦ (x)
9
which we evaluate by LattE and Ehrhart reciprocity, Equation (1):
x8
x8
=8
−
(1 − x)2 (1 − x2 )(1 − x4 ) (1 − x)2 (1 − x6 )
=
8x10 (2x2 + 1)
(1 − x)2 (1 − x4 )(1 − x6 )
8x10 (2x2 + 1)(x4 − x2 + 1)(x11 + x10 + · · · + x + 1)2 (x10 + x8 + · · · + x2 + 1)
=
.
(1 − x12 )4
From this generating function we extract the quasipolynomial
3
t −16t2 +76t−96

= (t−2)(t−6)(t−8)
,
if t ≡ 0, 2, 6, 8 (mod 12);

6
6



2

t3 −16t2 +73t−58


= (t−1)(t −15t+58)
, if t ≡ 1 (mod 12);

6
6



2

 t3 −16t2 +73t−102 = (t−3)(t −13t+34) , if t ≡ 3, 11 (mod 12);
6
6
Mc (t) =
3 −16t2 +76t−112
(t−4)(t2 −12t+28)
t


=
, if t ≡ 4, 10 (mod 12);

6
6




t3 −16t2 +73t−90

= (t−2)(t−5)(t−9)
,
if t ≡ 5, 9 (mod 12);

6
6




 t3 −16t2 +73t−70 = (t−7)(t2 −9t+10) ,
if t ≡ 7 (mod 12);
6
6
(8)
and the first few nonzero values for t > 0:
t
Mc (t)
mc (t)
10 11 12 13 14 15 16 17 18 19 20 21 22 23
24
8 16 40 64 96 128 184 240 320 400 504 608 744 880 1056
1 2 5 8 12 16 23 30 40 50 63 76 93 110 132
The last row is the number of symmetry classes, or normal squares, i.e., Mc (t)/8. The rows
are sequences A108576 and A108577 in the OEIS [15].
The symmetry of the constituents about residue 1 is curious.
The principal constituent is
t3 − 16t2 + 76t − 96
(t − 2)(t − 6)(t − 8)
=
.
6
6
Its unsigned constant term, 16, is the number of linear orderings of the cells that are induced
by magic squares. Thus, up to the symmetries of a magic square, there are just two order
types, even allowing arbitrarily large cell values. (The order types are illustrated in Example
3.11 of our general magic article, [6].)
We confirmed these results by direct enumeration, counting the strongly magic squares
for t ≤ 60 [7].
Compare this quasipolynomial to the weak quasipolynomial:
3 2
 t −3t +5t−3 = (t−1)(t2 −2t+3) , if t is odd;
6
6
 t3 −3t2 +8t−6 =
6
(t−1)(t2 −2t+6)
6
10
, if t is even;
with generating function
3.1.1
(x2 + 2x − 1)(2x3 − x2 + 1)
.
(1 − x)2 (1 − x2 )2
Reduced magic squares
A more fundamental count than the number of magic squares with an upper bound is the
number of reduced squares. Let Rmc (t) be the number of 3 × 3 reduced magic squares with
maximum cell value t, and rmc (t) the number of reduced symmetry types, or equivalently of
normalized reduced squares with maximum t. Then we have the formulas
Mc (t) =
t−1
X
(t − 1 − k)Rmc (k)
and
mc (t) =
k=0
t−1
X
(t − 1 − k)rmc (k),
k=0
since every reduced square with maximum k gives t−1−k unreduced squares with maximum
< t (and positive entries) by adding l to each entry where 1 ≤ l ≤ t − 1 − k. In terms of
generating functions,
Mc (x) =
x2
Rmc (x)
(1 − x)2
and
mc (x) =
x2
rmc (x).
(1 − x)2
(9)
We deduce the generating functions
rmc (x) =
x8 (2x2 + 1)
(1 − x4 )(1 − x6 )
and Rmc (x) = 8rmc (x), and from the latter the quasipolynomial



2t − 16, if t ≡ 0








2t
−
4,
if
t
≡
2,
10


mod 12;
Rmc (t) = 2t − 8, if t ≡ 4, 8 






2t − 12, if t ≡ 6



 0,
if t is odd
(10)
(1/8-th of these for rmc (t)) as well as the first few nonzero values:
t
Rmc (t)
rmc (t)
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
8 16 8 24 24 24 32 40 32 48 48 48 56 64 56
1 2 1 3 3 3 4 5 4 6 6 6 7 8 7
38 40 42
72 72 72
9 9 9
The sequences are A174256 and A174257 in the OEIS [15].
The principal constituent is 2t − 16, whose constant term in absolute value, 16, is the
number of linear orderings of the cells that are induced by magic squares—necessarily, the
same number as with Mc (t).
We confirmed the constituents by testing them against the coefficients of the generating
function for several periods.
11
Our way of reasoning, from all squares to reduced squares, is backward; logically, one
should count reduced squares and then deduce the ordinary magic square numbers from them
via Equation (9). Counting magic squares is not hard enough to require that approach, but
in treating semimagic and magilatin squares we follow the logical progression since then
reduced squares are much easier to handle.
3.2
Magic squares: Affine count (by magic sum)
The number of magic squares with magic sum t = 3γ is Ma (t). We take the normalized
coordinates x = α/t and y = β/t. The inside-out polytope is
Pa : 0 ≤ y ≤ x, x + y ≤ 13 ,
Ha : {h} where h : x = 2y.
The semilattice of flats is L(Pa◦ , Ha ) = {Pa◦ , h ∩ Pa◦ } with Pa◦ < h ∩ Pa◦ . The vertices are
O = (0, 0),
D = ( 31 , 0),
E = ( 92 , 19 ),
F = ( 16 , 61 ),
of which O, D, F are the vertices of Pa and O, E are the vertices of h ∩ Pa . From Equations
(1)–(3),
Ma (x) = 8E◦Pa◦ ,Ha (x) = 8 EPa◦ (x) − Eh∩Pa◦ (x)
x12
x12
−
=8
(1 − x3 )2 (1 − x6 ) (1 − x3 )(1 − x9 )
=
8x15 (2x3 + 1)
(1 − x3 )(1 − x6 )(1 − x9 )
=
8x15 (2x3 + 1)(x9 + 1)(x12 + x6 + 1)(x15 + x12 + · · · + x3 + 1)
.
(1 − x18 )3
From this generating function we extract the quasipolynomial
 2
2t −32t+144

= 92 (t2 − 16t + 72), if t ≡ 0 (mod 18);

9




2t2 −32t+78

= 92 (t − 3)(t − 13), if t ≡ 3 (mod 18);


9




 2t2 −32t+120
= 92 (t − 6)(t − 10), if t ≡ 6 (mod 18);

9


2
Ma (t) = 2t −32t+126
= 92 (t − 7)(t − 9),
if t ≡ 9 (mod 18);
9




2t2 −32t+96

= 92 (t − 4)(t − 12), if t ≡ 12 (mod 18);

9



2


= 92 (t2 − 16t + 51), if t ≡ 15 (mod 18);
 2t −32t+102

9



0,
if t 6≡ 0 (mod 3);
and the first few nonzero values for t > 0:
t
Ma (t)
ma (t)
15 18 21 24 27 30 33 36 39 42 45 48 51 54
8 24 32 56 80 104 136 176 208 256 304 352 408 472
1 3 4 7 10 13 17 22 26 32 38 44 51 59
12
(11)
The last row is the number of symmetry classes, or normalized squares, which is Ma (t)/8.
The two sequences are A108578 and A108579 in the OEIS [15].
The principal constituent is
2t2 − 32t + 144
2
= (t2 − 16t + 72),
9
9
whose constant term, 16, is the number of linear orderings of the cells that are induced by
magic squares—the same number as with Mc (t).
We verified our results by direct enumeration, counting the strong magic squares for
t ≤ 72 [7].
Compare the magic-square quasipolynomial to the weak quasipolynomial:
( 2
2t −6t+9
, if t ≡ 0 (mod 3);
9
0,
if t 6≡ 0 (mod 3);
due to MacMahon [12, Vol. II, par. 409, p. 163], with generating function
5x6 − 2x3 + 1
.
(1 − x3 )3
3.2.1
Reduced magic squares
Let Rma (t) be the number of 3 × 3 reduced magic squares with magic sum t, and rma (t)
the number of reduced symmetry types, or equivalently of normalized reduced squares with
magic sum t. Then
X
X
rma (s),
Rma (s) and ma (t) =
Ma (t) =
0<s<t
s≡t (mod 3)
0<s<t
s≡t (mod 3)
since every reduced square with sum s = t − 3k, where 0 < 3k ≤ t − 3, gives one unreduced
square with sum t (and positive entries) by adding 3k to each entry. In terms of generating
functions,
x3
ma (x) =
rma (x) ;
1 − x3
thus,
x12 (2x3 + 1)
rma (x) =
;
(1 − x6 )(1 − x9 )
and Rma (x) = 8rma (x). The quasipolynomial is


4


t − 16 = 43 (t − 12), if t ≡ 0


3








4
4


t
−
4
=
(t
−
3),
if
t
≡
3,
15

3
3



mod 18;
4
4


t
−
8
=
(t
−
6),
if
t
≡
6,
12
(12)
Rma (t) = 8rma (t) =

3
3







4

t − 12 = 43 (t − 9), if t ≡ 9


3




 0,
if t 6≡ 0
mod 18.
13
The initial nonzero values:
t
Rma (t)
rma (t)
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63
8 16 8 24 24 24 32 40 32 48 48 48 56 64 56 72 72 72
1 2 1 3 3 3 4 5 4 6 6 6 7 8 7 9 9 9
These sequences are A174256 and A174257 in the OEIS [15].
One of the remarkable properties of magic squares of order 3 is that Rmc (2k) = Rma (3k).
The reason is that the middle term of a reduced 3 × 3 magic square equals s/3, if s is the
magic sum, and the largest entry is 2s/3. Thus, the reduced squares of cubic and affine
type, allowing for the difference in parameters, are the same, and although the counts of
magic squares by magic sum and by upper bound differ, the only reason is that the reduced
squares are adjusted differently to get all squares.
The principal constituent 43 t − 16 has constant term whose absolute value is the same as
with all other reduced magic quasipolynomials.
We confirmed the constituents by comparing their values to the coefficients of the generating function for several periods.
4
Semimagic squares of order 3
Now we apply our approach to counting semimagic squares. Here is the general form of a
reduced, normalized 3 × 3 semimagic square, in which the magic sum is s = 2α + 2β + γ:
0
β
2α + β + γ
α+β
α+β+γ−δ
δ
α+β+γ
α+δ
β−δ
(13)
Proposition 1. A reduced and normalized 3 × 3 semimagic square has the form (13) with
the restrictions
0 < α, β, γ;
(14)
0 < δ < β;
and

β+γ
β−α β

 2 , 2, 2 ,
δ 6= β − α, α + γ;


γ.
14
β+α+γ
;
2
(15)
The largest entry in the square is w := x13 = 2α + β + γ.
Each reduced normal square with largest entry w corresponds to exactly 72(t − w − 1)
different magic squares with entries in the range (0, t), for 0 < w < t. Each reduced normal
square with magic sum s corresponds to exactly 72 different magic squares with magic sum
equal to t, if t ≡ s (mod 3), and none otherwise, for 0 < s < t.
Proof. By permuting rows and columns we can arrange that x11 = min xij and that the top
row and left column are increasing. By flipping the square over the main diagonal we can
further force x21 > x12 . By subtracting the least entry from every entry we ensure that
x11 = 0. Thus we account for the 72(t − w − 1) semimagic squares that correspond to each
reduced normal square.
The form of the top and left sides in (13) is explained by the fact that x11 < x12 <
x21 < x31 < x13 . The conditions xij > x11 for i, j = 2, 3, together with the row-sum and
column-sum equations, imply that x13 is the largest entry and that x23 < x12 .
The only possible equalities amongst the entries are ruled out by the following inequations:
x22 =
6 x12 , x21 , x23 ;
x32 =
6 x12 , x22 ;
x33 =
6 x23 , x32 .
These correspond to the restrictions (15):
x22 =
6 x12 ⇐⇒ δ =
6 α + γ;
x22 =
6 x21 ⇐⇒ δ =
6 γ;
β+α+γ
;
x22 6= x23 ⇐⇒ δ 6=
2
x32 6= x12 ⇐⇒ δ 6= β − α;
x32 6= x22 ⇐⇒ δ 6= β + γ − δ ⇐⇒ δ 6=
x33 6= x23 ⇐⇒ δ 6= β − δ ⇐⇒ δ 6=
β+γ
;
2
β
;
2
x33 6= x32 ⇐⇒ δ 6= α + δ 6= β − δ ⇐⇒ δ 6=
4.1
β−α
.
2
Semimagic squares: Cubical count (by upper bound)
We are counting squares by a strict upper bound on the allowed value of an entry; this bound
is the parameter t. Let Sc (t), for t > 0, be the number of semimagic squares of order 3 in
which every entry belongs to the range (0, t).
4.1.1
Counting the weak squares
The polytope Pc is the 5-dimensional intersection of [0, 1]9 with the semimagic subspace in
which all row and column sums are equal. This polytope is integral because it is the intersection with an integral polytope of a subspace whose constraint matrix is totally unimodular;
15
so the quasipolynomial is a polynomial. By contrast, the inside-out polytope (P, H) for
enumerating strong semimagic squares has denominator 60. We verified by computer counts
for t ≤ 18 that the weak polynomial is
3t5 − 15t4 + 35t3 − 45t2 + 32t − 10
(t − 1)(t2 − 2t + 2)(3t2 − 6t + 5)
=
10
10
with generating function
(7x2 − 2x + 1)(2x3 + x2 + 4x − 1)
.
(1 − x)6
We conclude that Pc has volume 3/10.
4.1.2
Reduction of the number of strong squares
We compute Sc (t) via Rc (w), the number of reduced squares with largest entry equal to w.
The formula is
t−1
X
Sc (t) =
(t − 1 − w)Rc (w).
(16)
w=0
The value Rc (w) = 72rc (w) where rc (w) is the number of normal reduced squares (in which
we know the largest entry to be x13 ). Thus rc (s) counts the number of 1s -integral points in
the interior of the 3-dimensional polytope Qc defined by
0 ≤ x, y;
0 ≤ z ≤ y;
x+y ≤
1
2
(17)
with the seven excluded (hyper)planes
z=
y − x y 1 − y − 2x 1 − x − y
, ,
,
, y − x, 1 − x − 2y, 1 − 2x − 2y ,
2
2
2
2
(18)
the three coordinates being x = α/w, y = β/w, and z = δ/w.
The hyperplane arrangement for reduced normal squares is that of (18). We call it Ic .
Thus rc (s) = EQ◦ ◦c ,Ic (s).
4.1.3
Geometrical analysis of the reduced normal polytope
We apply Möbius inversion, Equation (3), over the intersection poset L(Q◦c , Ic ). We need
to know not only L(Q◦c , Ic ) but also all the vertices of (Qc , Ic ), since they are required for
computing the Ehrhart generating function and estimating the period of the reduced normal
quasipolynomial.
16
We number the planes:
π1
π2
π3
π4
π5
π6
π7
: x − y + 2z = 0,
:
y − 2z = 0,
:
2x + 2z = 1,
:
x + 2z = 1,
: x − y + z = 0,
: x + y + z = 1,
: 2x + y + z = 1.
The intersection of two planes, πj ∩ πk , is a line we call ljk ; π3 ∩ π5 ∩ π6 is a line we also
call l356 . The intersection of three planes is, in general, a point but not usually a vertex of
(Qc , Ic ).
Our notation for the line segment with endpoints X, Y is XY , while XY denotes the
entire line spanned by the points. The triangular convex hull of three noncollinear points
X, Y, Z is XY Z. We do not need quadrilaterals, as the intersection of each plane with Q is
a triangle.
We need to find the intersections of the planes with Q◦c , separately and in combination.
Here is a list of significant points; we shall see it is the list of vertices of (Qc , Ic ). The first
column has the vertices of Qc , the second the vertices of (Qc , Ic ) that lie in open edges, the
third the vertices that lie in open facets, and the last is the sole interior vertex.
O = (0, 0, 0),
A = ( 12 , 0, 0),
Bc = (0, 1, 0),
Cc = (0, 1, 1),
Dc
Ec
Ec′
Ec′′
= (0, 21 , 21 ) ∈ OC,
= ( 31 , 13 , 0) ∈ AB,
= ( 31 , 13 , 31 ) ∈ AC,
= (0, 1, 12 ) ∈ BC,
Fc
Gc
G′c
G′′c
= (0, 23 , 13 ) ∈ OBC,
= ( 14 , 21 , 41 ) ∈ ABC,
= ( 15 , 53 , 51 ) ∈ ABC,
= ( 15 , 53 , 52 ) ∈ ABC,
Hc = ( 51 , 25 , 51 ).
(19)
The denominator of (Qc , Ic ) is the least common denominator of all the points; it evidently
equals 4 · 3 · 5 = 60.
The intersections of the planes with the edges of Qc are in Table 1. The subscript c is
omitted.
Table 2 shows the lines generated by pairwise intersection of planes.
17
Plane
Intersection with edge
Intersection
OA
OB
OC
AB
AC
BC
with Q
π1
O
O
O
E
∈
/Q
E ′′
OEE ′′
π2
OA
O
O
A
A
E ′′
OAE ′′
π3
A
∅
D
A
A
E ′′
ADE ′′
π4
∈
/Q
∅
D
∈
/Q
E′
E ′′
DE ′′ E ′
π5
O
O
OC
E
C
C
OCE
π6
∈
/Q
B
D
B
E′
B
BDE ′
π7
A
B
D
AB
A
B
ABD
Table 1: Intersections of planes of I with edges of Q. A vertex of Q contained in πj will show
up three times in the row of πj . In order to clarify the geometry, we distinguish between a
plane’s meeting an edge line outside Q and not meeting it at all (i.e., their being parallel).
π1
π2
π3
π2
π3
π4
π5
x=0
y = 2z
x + z = 12
y − z = 21
y = 2z
x + z = 12
y =1
x + 2z = 1
y = 2z
x+y =1
x=0
z = 12
z =0
x=y
y = 2z
x=z
l356 :
x = 1 − 2z
y =1−z
π4
π5
π6
z = 2y − 1
x = 2 − 3y
y = 2z
x + 3z = 1
x + z = 12
y = 21
x = 1 − 2z
y =z
l356
π6
π7
x + z = 13
y − z = 31
y = 2z
2x + 3z = 1
x + z = 12
y =z
x = 1 − 2z
y = 3z − 1
x = 1 − 2y
z = 3y − 1
x=0
y+z =1
Table 2: The equations of the pairwise intersections of planes of Ic .
In Table 3 we describe the intersection of each line with Qc and with its interior. The
subscript c is omitted.
18
π2
′′
π1
π2
π3
OE
(OBC)
π3
π4
′′
′′
E
(BC)
AE ′′
(ABC)
E
(BC)
E ′′
(BC)
DE ′′
(OBC)
π5
π6
π7
OE
(OAB)
F G′
EF
OG
FG
AF
DG′′
π4
DE ′
(OAC)
AD
(OAC)
D
(OC)
l356
ED
l356 : DG
π5
BD
(OBC)
π6
Table 3: The intersections of lines with Q and Q◦ . The second (parenthesized) row in each
box shows the smallest face of Q to which the intersection belongs, if that is not Q itself;
these intersections are not part of the intersection poset of (Q◦ , I).
Last, we need the intersection points of three planes of Ic ; or, of a plane and a line. Some
are not in Qc at all; them we can ignore. Some are on the boundary of Qc ; they are necessary
in finding the denominator, but all of them are points already listed in (19). It turns out
that
π2 ∩ π5 ∩ π7 = Hc
is the only vertex in Q◦c , so it is the only one we need for the intersection poset.
Here, then, is the intersection poset (Figure 1). The subscript c is omitted. In the figure,
for simplicity, we write πj , etc., when the actual element is the simplex πj ∩ Q◦ , etc.; we also
state the vertices of the simplex. The Möbius function µ(0̂, u) equals (−1)codim u with the
exception of µ(0̂, l356 ) = 2.
4.1.4
Generating functions and the quasipolynomial
◦
That was the first half of the work. The second half begins with finding rc (w) = E(Q
◦ ,I ) (w)
c c
from the Ehrhart generating functions Eu (t) of the intersections by means of (6). The next
step, then, is to calculate all necessary generating functions. This is done by LattE. rc (x)
is the result of applying reciprocity to the sum of all these rational functions (with the
appropriate Möbius-function multiplier −1, excepting EDG◦ (x) whose multiplier is −2). The
19
H
l45 (DG′′ ) l16 (F G′ ) l17 (EF )
π4 (DE ′′ E ′ )
π1 (OEE ′′ )
l356 (DG)
π6 (BDE ′ )
l26 (F G)
π5 (OCE)
l57 (ED) l25 (OG) l27 (AF )
π3 (ADE ′′ )
π7 (ABD)
π2 (OAE ′′ )
R3 (OABC)
Figure 1: The intersection poset L(Q◦ , I) for semimagic squares. The diagram shows both
the flats and (in parentheses) their intersections with Q.
result is:
−rc (1/x) = EOABC (x) + EOEE ′′ (x) + EOAE ′′ (x)
+ EADE ′′ (x) + EDE ′ E ′′ (x) + EOCE (x)
+ EBDE ′ (x) + EABD (x) + EF G′ (x)
+ EEF (x) + EOG (x) + EF G (x)
+ EAF (x) + 2EDG (x) + EDG′′ (x)
+ EDE (x) + EH (x)
1
1
1
+
+
=
3
2
2
3
(1 − x) (1 − x ) (1 − x)(1 − x )(1 − x ) (1 − x)(1 − x2 )2
1
1
1
+
+
+
(1 − x2 )3 (1 − x2 )2 (1 − x3 ) (1 − x)2 (1 − x3 )
1
1
1
+
+
+
2
3
2
2
3
(1 − x)(1 − x )(1 − x ) (1 − x)(1 − x )
(1 − x )(1 − x5 )
1
1
1
+
+
+
3
2
4
3
(1 − x )
(1 − x)(1 − x ) (1 − x )(1 − x4 )
1
1
1
+2
+
+
2
3
2
4
2
(1 − x )(1 − x )
(1 − x )(1 − x ) (1 − x )(1 − x5 )
1
1
+
.
+
(1 − x2 )(1 − x3 ) 1 − x5
20
(20)
Then by (16) the generating function for cubically counted semimagic squares is
Sc (x) = 72
x2
rc (x)
(1 − x)2
72x10 [18 x9 + 46 x8 + 69 x7 + 74 x6 + 65 x5 + 46 x4 + 26 x3 + 11 x2 + 4 x + 1]
.
(1 − x2 )2 (1 − x3 )2 (1 − x4 )(1 − x5 )
(21)
From the geometrical denominator 60 or the (standard-form) algebraic denominator (1 −
x60 )5 we know the period of Sc (t) divides 60. We compute the constituents of Sc (t) by the
method of Section 2.1; the result is that

3 5 75 4 331 3 5989 2


t − t +
t −
t + c1 (t)t − c0 (t), if t is even;

 10
8
3
10
Sc (t) =
(22)


75
331
11933
3

 t5 − t4 +
t3 −
t2 + c1 (t)t − c0 (t), if t is odd;
10
8
3
20
=
where c1 varies with period 6, given by

1464,



1456,
c1 (t) = 2831

,

2

 2847
,
2
if
if
if
if
t ≡ 0, 2
t≡4
t≡1
t ≡ 3, 5
(mod 6);
and c0 , given by Table 4, varies with period 60. (It is curious that the even constant terms
have half the period of the odd terms.) Thus the period of Sc turns out to be 60, the largest
it could be.
21
t
c0 (t)
t
c0 (t)
t
c0 (t)
t
c0 (t)
t
c0 (t)
0
1296
12
1296
24
36
110413
120
3824
3
47727
40
18152
15
25705
24
6192
5
25193
24
19552
15
44847
40
3544
3
130253
120
13
120781
120
19552
15
9315
8
16856
15
25705
24
6624
5
129421
120
3824
3
41391
40
3544
3
140621
120
25
29
41
6192
5
23465
24
19552
15
47727
40
3544
3
121613
120
48
1
6624
5
23465
24
18256
15
9315
8
18152
15
131981
120
30
1296
42
1296
54
31
119053
120
3824
3
44847
40
18152
15
27433
24
43
129421
120
19552
15
8739
8
16856
15
27433
24
55
6624
5
120781
120
3824
3
44271
40
3544
3
131981
120
6624
5
25193
24
18256
15
8739
8
18152
15
140621
120
2
3
4
5
6
7
8
9
10
11
14
15
16
17
18
19
20
21
22
23
26
27
28
32
33
34
35
37
38
39
40
44
45
46
47
49
50
51
52
53
56
57
58
59
Table 4: Constant terms (without the negative sign) of the constituents of the semimagic
cubical quasipolynomial Sc (t).
The principal constituent (for t ≡ 0) is
3 5 75 4 331 3 5989 2
t − t +
t −
t + 1464t − 1296.
10
8
3
10
Its unsigned constant term, 1296, is the number of order types of semimagic squares. Allowing for the 72 symmetries of a semimagic square, there are just 18 symmetry classes of
order types.
For the first few nonzero values of Sc (t) see the following table. (This is sequence A173546
in the OEIS [15].) The third row is the number of normalized squares, or symmetry classes
(sequence A173723), which equals Sc (t)/72. The other lines give the numbers of reduced
squares (sequence A173727) and of reduced normal squares (i.e., symmetry types of reduced
squares; sequence A173724), which may be of interest.
t
Sc (t)
sc (t)
Rc (t)
rc (t)
8
9
10
11
12
13
14
15
16
17
18
19
0
0
72 288 936 2592 5760 11520 20952 35712 57168 88272
0
0
1
4
13
36
80
160
291
496
794
1226
72 144 432 1008 1512 2592 3672 5328 6696 9648 11736 15552
1
2
6
14
21
36
51
74
93
134
163
216
Compare the strong to the weak quasipolynomial. The leading coefficients agree and
the strong coefficient of t4 is constant. These facts, of which we made no use in deducing
22
the quasipolynomial, provide additional verification of the correctness of the counts and
constituents.
4.1.5
Another method: Direct counting
We checked the constituents by directly counting (in Maple) all semimagic squares for
t ≤ 100. The numbers agreed with those derived from the generating function and quasipolynomial above.
4.2
Semimagic squares: Affine count (by magic sum)
Now we count squares by magic sum: we compute Sa (t), the number of squares with magic
sum t.
4.2.1
The Birkhoff polytope
The polytope P for semimagic squares of order 3, counted by magic sum, is 4-dimensional
and integral. (It is the polytope of doubly stochastic matrices of order 3, i.e., a Birkhoff
polytope [8, 4].)
4.2.2
Affine weak semimagic
The polytope for weak semimagic squares of order 3 is the same P .
The weak quasipolynomial, or rather, polynomial, first computed by MacMahon [12, Vol.
II, par. 407, p. 161], is
t4 − 6t3 + 15t2 − 18t + 8
(t − 1)(t − 2)(t2 − 3t + 4)
=
8
8
with generating function
4.2.3
6x4 − 9x3 + 10x2 − 5x + 1
.
(1 − x)5
Reduction
The count is via Ra (s), the number of reduced squares with magic sum s. The formula is
X
Ra (s)
if t > 0.
(23)
Sa (t) =
0<s≤t−3
s≡t (mod 3)
We have Ra (s) = 72ra (w), where ra (s) is the number of reduced, normalized squares with
magic sum s, equivalently the number of 1s -integral points in the interior of the 3-dimensional
polytope Qa defined by
0 ≤ x, y;
0 ≤ z ≤ y;
23
x+y ≤
1
2
(24)
with the seven excluded (hyper)planes
z=
y − x y 1 − y − 2x 1 − x − y
, ,
,
, y − x, 1 − x − 2y, 1 − 2x − 2y ,
2
2
2
2
(25)
the three coordinates being x = α/s, y = β/s, and z = δ/s.
The hyperplane arrangement for reduced, normalized squares is that of (25). We call it
Ia . Thus ra (s) = EQ◦ ◦a ,Ia (s).
4.2.4
The reduced, normalized weak polytopal quasipolynomial
This function simply counts 1s -lattice points in Q◦a . The counting formula is
summed over all triples that satisfy (14). It simplifies to
X ⌊ s−1 ⌋ − α
2
,
2
α
which gives the Ehrhart quasipolynomial
 s−1 
 2
=

s−1 
3

⌊ 2 ⌋
EQ◦a (s) =
=
s−2 
3



 2
=
3
1
(s
48
P P P
δ
1,
(26)
1
(s
48
− 2)(s − 4)(s − 6), for even s.
EP ◦ (x) =
x7 (1 + x)
(1 − x2 )4
EP (x) =
1+x
.
(1 − x2 )4
and by reciprocity that
Geometrical analysis of the reduced, normalized polytope
We apply Möbius inversion, Equation (6), over the intersection poset L(Q◦a , I).
We number the planes:
π1
π2
π3
π4
π5
π6
π7
β
− 1)(s − 3)(s − 5), for odd s;
The leading coefficient is vol Qa .
We deduce from (26) that
4.2.5
α
: x − y + 2z
:
y − 2z
: 2x + y + 2z
: x + y + 2z
:
x−y+z
: x + 2y + z
: 2x + 2y + z
24
= 0,
= 0,
= 1,
= 1,
= 0,
= 1,
= 1.
The intersection of two planes, πj ∩ πk , is a line we call ljk ; π3 ∩ π5 ∩ π6 is a line we also
call l356 . The intersection of three planes is, in general, a point but not usually a vertex of
(Qa , Ia ). Our geometrical notation is as in the cubical analysis.
We need to find the intersections of the planes with Q◦a , separately and in combination.
Here is a list of significant points; we shall see it is the list of vertices of (Qa , Ia ). The first
column has the vertices of Qa , the second the vertices of (Qa , Ia ) that lie in open edges, the
third the vertices that lie in open facets, and the last is the sole interior vertex.
O = (0, 0, 0),
A = ( 21 , 0, 0),
Ba = (0, 21 , 0),
Ca = (0, 21 , 21 ),
Da
Ea
Ea′
Ea′′
= (0, 13 , 13 ) ∈ OC,
= ( 14 , 41 , 0) ∈ AB,
= ( 41 , 41 , 41 ) ∈ AC,
= (0, 21 , 14 ) ∈ BC,
Fa
Ga
G′a
G′′a
= (0, 25 , 51 ) ∈ OBC,
= ( 61 , 62 , 16 ) ∈ ABC,
= ( 18 , 83 , 18 ) ∈ ABC,
= ( 81 , 83 , 28 ) ∈ ABC.
Ha = ( 71 , 72 , 17 ),
(27)
The least common denominator of O, A, Ba , Ca explains the period 2 of EQ◦a . The denominator of (Qa , Ia ) is the least common denominator of all the points; it evidently equals
8 · 3 · 5 · 7 = 840.
The intersections of the planes with the edges of Qa are in Table 1. Table 5 shows the
lines generated by pairwise intersection of planes. Table 3 describes the intersection of each
line with Qa and with Q◦a .
π2
π1
π2
π3
π3
x=0
x=
y = 2z
y=
x+y
1−4z
3
1+2z
3
= 21
y = 2z
π4
π5
π6
x=y
x = 2 − 5y
x=
z =0
z = 3y − 1
y=
x = 1 − 2y
x=z
x = 1 − 5z
x=
y = 2z
y = 2z
y = 2z
y = 2z
x + 2z =
y=
1
2
1
2
x=0
y = 1 − 2z
π4
π5
l356 :
x+z =
y=
1
3
1
3
x = 3y − 1
x = 1 − 3y
z = 1 − 2y
z =y
l356
π7
1−5z
4
1+3z
4
1−5z
2
2x + 3y = 1
z =y
x+y =
z=
1
3
1
3
x = 1 − 3y
z = 4y − 1
x=0
π6
z = 1 − 2y
Table 5: The equations of the pairwise intersections of planes of Ia .
Last, we need the intersection points of three planes of Ia ; or, of a plane and a line. Some
are not in Qa at all; them we can ignore. Some are on the boundary of Qa ; they are necessary
in finding the denominator, but all of them are points already listed in (27). It turns out
25
that
π2 ∩ π5 ∩ π7 = Ha
is the only vertex in Q◦a , so it is the only one we need for the intersection poset.
The combinatorial structure and the intersection poset (Figure 1) for the affine count
are identical to those for the cubical count. The reason is that the affine polytope Pa is
the 4-dimensional section of Pc by the flat in which the magic sum equals 1, and this flat is
orthogonal to the line of intersection of the whole arrangement Ha .
4.2.6
Generating functions and the quasipolynomial
The second half of the affine solution is to find ra (s) = E◦(Q◦a ,Ia ) (s) by applying Equations (1)–
(4) after finding the Ehrhart generating functions Eu (s) for u ∈ L(Q◦a , Ia ). The next step,
then, is to calculate those generating functions. This is done by LattE. Then (−1)3 ra (x−1 )
is the sum of all these rational functions; that is,
−ra (1/x) = EOABC (x) + EOEE ′′ (x) + EOAE ′′ (x)
+ EADE ′′ (x) + EDE ′′ E ′ (x) + EOCE (x)
+ EBDE ′ (x) + EABD (x) + EF G′ (x)
+ EEF (x) + EOG (x) + EF G (x)
+ EAF (x) + 2EDG (x) + EDG′′ (x)
+ EDE (x) + EH (x)
1
1
1
=
+
+
2
3
4
2
(1 − x)(1 − x )
(1 − x)(1 − x )
(1 − x)(1 − x2 )(1 − x4 )
1
1
1
+
+
+
2
3
4
3
4
2
(1 − x )(1 − x )(1 − x ) (1 − x )(1 − x )
(1 − x)(1 − x2 )(1 − x4 )
1
1
1
+
+
+
(1 − x2 )(1 − x3 )(1 − x4 ) (1 − x2 )2 (1 − x3 ) (1 − x5 )(1 − x8 )
1
1
1
+
+
+
4
5
6
5
(1 − x )(1 − x ) (1 − x)(1 − x ) (1 − x )(1 − x6 )
1
1
1
+2
+
+
2
5
3
6
3
(1 − x )(1 − x )
(1 − x )(1 − x ) (1 − x )(1 − x8 )
1
1
+
+
.
3
4
(1 − x )(1 − x ) 1 − x7
(28)
The generating function for the affine count of semimagic squares, by (23), is
Sa (x) = 72
72x15
(
x3
ra (x) =
1 − x3
18x21 + 5x20 + 15x19 + 11x17 − 8x16 + x15 − 23x14 − 13x13 − 22x12 − 9x11
− 16x10 + x9 − 3x8 + 7x7 + 7x6 + 9x5 + 7x4 + 6x3 + 4x2 + 2x + 1
(1 − x3 )2 (1 − x4 )(1 − x5 )(1 − x6 )(1 − x7 )(1 − x8 )
)
.
(29)
26
From the geometrical or generating-function denominator we know that the period of
Sa (t) divides 840 = lcm(3, 4, 6, 7, 8). This is long, but it can be simplified. The factor 7 in
the period is due to a single term in (28). If we treat it separately we have ra as a sum
x7
of the H-term x7 /(1 − x7 ) and a “truncated” generating function for ra (x) + 1−x
7 , and a
corresponding truncated expression
x10
=
(1 − x3 )(1 − x7 )
)
(
17x19 + 5x18 + 15x17 + x16 + 12x15 − 7x14 + 2x13 − 7x12
− 8x11 − 9x10 − 9x9 − 6x8 − 6x7 − x6 + x4 + x3 − 1
Sa (x) − 72
−72x10
(1 − x3 )2 (1 − x4 )(1 − x5 )(1 − x6 )(1 − x8 )
.
We extract the constituents from this expression as in Section 2.1, separately for the two
parts of the generating function. The constituents are all of the form
9
1
Sa (t) = t4 − t3 + a2 (t)t2 − a1 (t)t + a0 (t) − 72S7 (t),
8
2
(30)
where S7 (t) is a correction, to be defined in a moment, and
 243
, if t ≡ 0

4


 218 , if t ≡ 1, 5
4
(mod 6);
a2 (t) = 227

,
if
t
≡
2,
4

4

 234
, if t ≡ 3
4

1968

, if t ≡ 0

5


1158

, if t ≡ 1, 5


5


1383

, if t ≡ 2, 10

5


 1653 , if t ≡ 3
5
(mod 12);
a1 (t) = 1428

,
if
t
≡
4,
8

5


1923


, if t ≡ 6

5


1113


, if t ≡ 7, 11

5

 1698
, if t ≡ 9
5
and a0 (t) is given in Table 6.
We call the constituents of the quasipolynomial
9
1
Sa (t) + 72S7 (t) = t4 − t3 + a2 (t)t2 − a1 (t)t + a0 (t)
8
2
the truncated constituents of Sa (t), since they correspond to the truncated generating function mentioned just above. The S7 term that undoes the truncation is
(
1, if t ≡ 10, 13, 16, 17, 19, 20 (mod 21);
t−1
+
S7 (t) :=
21
0, otherwise
27
t − t̄
+ s7 (t),
21
where t̄ := the least positive residue of t modulo 7 and
(
1, if t ≡ 10, 13, 16, 17, 19, 20 (mod 21);
s7 (t) :=
0, otherwise.
=
Note that t̄ = 21 if t ≡ 0, so that S7 (0) = −1 and in general S7 (21k) = k − 1.
t a0 (t)
t a0 (t)
t a0 (t)
t a0 (t)
t a0 (t)
t a0 (t)
0
1224
20
524
40
584
60
1188
80
560
100
548
1
21
89
29979
40
1921
5
−613
40
5652
5
2431
8
1333
5
27963
40
2344
5
7451
40
101
69
8699
40
1621
5
24507
40
2632
5
1951
8
4833
5
3803
40
2044
5
32571
40
81
49
6299
40
5121
5
347
40
2332
5
6975
8
1453
5
4283
40
5544
5
8411
40
61
29
31419
40
1741
5
827
40
5832
5
2143
8
1513
5
26523
40
2164
5
8891
40
41
9
7259
40
1801
5
23067
40
2452
5
2239
8
4653
5
5243
40
2224
5
31131
40
109
7739
40
4941
5
1787
40
2512
5
6687
8
1633
5
2843
40
5364
5
9851
40
10
413
30
1017
50
389
70
377
90
1053
110
353
11
539
40
5796
5
7547
40
1477
5
5823
8
2488
5
8603
40
4689
5
2651
40
31
2939
40
2656
5
28827
40
1777
5
799
8
5508
5
11003
40
1189
5
26811
40
51
24219
40
2596
5
6587
40
4797
5
1279
8
2368
5
32283
40
1489
5
1691
40
71
1979
40
5976
5
6107
40
1657
5
5535
8
2308
5
10043
40
4509
5
4091
40
91
1499
40
2476
5
30267
40
1597
5
1087
8
5688
5
9563
40
1369
5
25371
40
111
25659
40
2776
5
5147
40
4977
5
991
8
2188
5
33723
40
1309
5
3131
40
2
3
4
5
6
7
8
12
13
14
15
16
17
18
19
22
23
24
25
26
27
28
32
33
34
35
36
37
38
39
42
43
44
45
46
47
48
52
53
54
55
56
57
58
59
62
63
64
65
66
67
68
72
73
74
75
76
77
78
79
82
83
84
85
86
87
88
92
93
94
95
96
97
98
99
102
103
104
105
106
107
108
112
113
114
115
116
117
118
119
Table 6: Constant terms of the truncated constituents of Sa (t).
The period of the constant term of the truncated constituents is 120. It follows that Sa (t)
has period 7 · 840, that is, 840.
28
The principal constituent of Sa (t) (that is, for t ≡ 0) is
1 4 9 3 243 2 13896
t − t +
t −
t + 1296.
8
2
4
35
(This incorporates the effect of the term −72S7 .) The constant term is the same as in the
cubic count, as it is the number of order types of semimagic squares.
We give the first few nonzero values of Sa (t) in the following table. (This sequence is
A173547 in the OEIS [15].) The third row is the number of normalized squares, or symmetry
classes (sequence A173725); this is Sa (t)/72. The last rows are the numbers of reduced
squares (sequence A173728) and of reduced, normalized squares (sequence A173726) with
magic sum t.
t
Sa (t)
sa (t)
Ra (t)
ra (t)
4.2.7
12 13 14 15 16
17
18
19
20
21
22
23
24
0
0
0
72 144 288 576 864 1440 2088 3024 3888 5904
0
0
0
1
2
4
8
12
20
29
42
54
82
72 144 288 504 720 1152 1512 2160 2448 3816 3960 5544 6264
1
2
4
7
10
16
21
30
34
53
55
77
87
Alternative methods: Direct counting and direct computation
We verified our formulas by computing Sa (t) for t ≤ 100 through direct enumeration of normal squares. The results agree with those computed by expanding the generating function.
We also applied Proposition 1 to derive a formula, independent of all other methods, by
which we calculated numbers (which we are not describing; see the “Six Little Squares” Web
page [7]) that allowed us to find the 840 constituents by interpolation. These interpolated
constituents fully agreed with the ones given above.
5
Magilatin squares of order 3
A magilatin square is like a semimagic square except that entries may be equal if they are in
different rows and columns. The inside-out polytope is the same as with semimagic squares
except that we omit those hyperplanes that prevent equality of entries in different rows and
columns. Thus, in our count of reduced squares, we have to count the fractional lattice
points in some of the faces of the polytope.
The reduced normal form of a magilatin square is the same as that of a semimagic square
except that the restrictions are weaker. It might be thought that this would introduce
ambiguity into the standard form because the minimum can occur in several cells, but it
turns out that it does not.
Proposition 2. A reduced, normal 3 × 3 magilatin square has the form (13) with the restrictions
0 < β, γ;
0 ≤ α;
(31)
0 ≤ δ ≤ β;
29
and (15). Each reduced square with w in the upper right corner corresponds to exactly
t − w − 1 different magilatin squares with entries in the range (0, t), for 0 < w < t. Each
reduced square with magic sum s corresponds to one magilatin square with magic sum equal
to t, if t ≡ s (mod 3), and none otherwise, for 0 < s < t.
Proof. The proof is similar to that for semimagic squares; we can arrange the square by
permuting rows and columns and by reflection in the main diagonal so that x11 is the
smallest entry, the first row and column are each increasing, and x21 ≥ x12 . We cannot say
x21 > x12 because entries that do not share a row or column may be equal. Still, we obtain
the form (13) with the bounds (31) and the same inequations (15) as in semimagic because
all the latter depend on having no two equal values in the same line (row or column).
Each reduced, normal magilatin square gives rise to a family of true magilatin squares
by adding a positive constant to each entry and by symmetries, which are generated by row
and column permutations and reflection in the main diagonal. Call the set of symmetries
G. As with semimagic, |G| = 2(3!)2 = 72. Each normal, reduced square S gives rise to
|G/GS | = 72/|GS | squares via symmetries, where GS is the stabilizer subgroup of S. If all
entries are distinct, then the square is semimagic, GS is trivial, and everything is as with
semimagic squares. However, if α = 0 or δ = 0 or δ = β, the stabilizer is nontrivial. We
consider each case in turn.
The case α = 0 < δ. Here δ < β because no line can repeat a value. To fix the square we
cannot permute any rows or columns but we can reflect in the main diagonal, so |GS | = 2.
Moreover, (15) reduces to
β β+γ
.
δ 6= γ, ,
2
2
We are in OBC, the x = 0 facet of Q, with the induced arrangement of three lines, Ix=0 .
The number of reduced magilatin squares of this kind is 36rOBC (t), where rOBC (t) is the
number of 1t -lattice points in the open facet and, equivalently, the number of symmetry types
of reduced magilatin squares of this kind. We apply Equations (1)–(4) to the intersection
poset L(OBC ◦ , Ix=0 ), which is found in Figure 2. The Möbius function µ(OBC, u) equals
(−1)codim u .
The case δ = 0 < α. In this case a nontrivial member of GS can only exchange the two
zero positions. Such a symmetry that preserves the increase of the first row and column is
unique (as one can easily see); thus |GS | = 2. Furthermore, (15) reduces to
α 6= β.
We are in OAB, the z = 0 facet, with the induced arrangement Iz=0 of one line. The number
of reduced magilatin squares of this kind is 36rOAB (t), where rOAB (t) is the number of 1t lattice points in the open facet and, equivalently, the number of symmetry types of reduced
squares. We apply Equations (1)–(4) to the intersection poset L(OAB ◦ , Iz=0 ), shown in
Figure 2. The Möbius function µ(OAB, u) equals (−1)codim u .
The case δ = β. Here we must have α > 0. There are two zero positions in opposite
corners. A symmetry that exchanges them and preserves increase in the first row and column
is uniquely determined, so |GS | = 2. The inequations reduce to
β 6= γ, α + γ.
30
We are in OAC, the facet where y = z, with the two-line induced arrangement Iy=z . The
number of reduced magilatin squares of this kind is 36rOAC (t), where rOAC (t) is the number
of 1t -lattice points in the open facet, equally the number of reduced symmetry types. We
apply Equations (1)–(4) to the intersection poset L(OAC ◦ , Iy=z ) in Figure 2. The Möbius
function µ(OAC, u) equals (−1)codim u .
The case α = 0 = δ. In these squares there are three zero positions and the whole
square is a cyclic latin square. Any symmetry that fixes the zero positions also fixes the rest
of the square. There are 3! symmetries that permute the zero positions, generated by row
and column permutations. They all preserve the entire square. Therefore |GS | = 6. The
inequations disappear. We are in the edge OB, which is the face where x = z = 0, with the
empty arrangement, Ix=z=0 = ∅. The number of reduced magilatin squares of this kind is
12rOB (t), where rOB (t) is the number of 1t -lattice points in the open edge, also the number
of reduced symmetry types. The intersection poset L(OB ◦ , ∅) consists of the one element
OB, whose Möbius function µ(OB, OB) = 1.
To get the intersection posets we may examine Tables 1 and 3 to find the edges and
vertices of (Q◦ , I) in each closed facet. We also need to know which vertex is in which edge;
this is easy. Although we do not need the fourth facet, ABC, we include it for the interest
of its more complicated geometry.
Of course all the functions rs and Rml depend on whether we are counting cubically or
affinely (thus, subscripted c or a); the two types will be treated separately. But the general
conclusions hold that
Rml (x) = 72rs (x) + 36[rOAB (x) + rOAC (x) + rOBC (x)] + 12rOB (x),
(32)
and for the number of reduced symmetry types, rml (t) with generating function rml (x),
rml (x) = rs (x) + rOAB (x) + rOAC (x) + rOBC (x) + rOB (x),
(33)
where rs (x) is from semimagic and, by Equation (4) since |µ(X, Y )| = 1 for every lower
interval in each facet poset (except facet ABC, which we do not use),
(−1)3 rOAB (1/x) = EOAB (x) + EOE (x),
(−1)3 rOAC (1/x) = EOAC (x) + EAD (x) + EDE ′ (x),
(−1)3 rOBC (1/x) = EOBC (x) + EOE ′′ (x) + EBD (x) + EDE ′′ (x) + EF (x),
(−1)2 rOB (1/x) = EOB (x);
(34)
the sign and reciprocal on the left result from Equation (6).
There is also the generating function of the number of cubical or affine symmetry classes,
l(t), whose generating function is l(x). This is obtained from rml (t) in the same way as L(t)
is from Rml (t), the exact way depending on whether the count is affine or cubic.
5.1
Magilatin squares: Cubical count (by upper bound)
The weak quasipolynomial is exactly as in the semimagic cubical problem.
31
F
OE
DE ′
AD
OE ′′
OAB
BD
DE ′′
OAC
OBC
L(OAB ◦ , Iz=0 )
L(OAC ◦ , Iy=z )
G′
L(ABC ◦ , Ix+y=1/2 )
EE ′′
L(OBC ◦ , Ix=0 )
G′′
G
BE ′
AE ′′
CE
E ′ E ′′
ABC
Figure 2: The four facet intersection posets of (Q◦ , I).
32
5.1.1
Magilatin squares by upper bound
The number of 3 × 3 magilatin squares with strict upper bound t is Lc (t). We count them
via Rmlc (w), the number of reduced magilatin squares with largest entry (which we know to
be x13 ) equal to w. The formula is
t−1
X
Lc (t) =
(t − 1 − w)Rmlc (w).
(35)
w=0
Equivalently, Rmlc (w) counts w1 -integral points in the interior and part of the boundary of
the inside-out polytope (Qc , Ic ) of Section 4.1, weighted variably by 72/|GS |.
Now we must calculate the closed Ehrhart generating function for each necessary face.
This is done by LattE; here are the results. First, OABc :
(−1)3 rOABc (1/x) = EOABc (x) + EOEc (x)
1
1
+
=
2
2
(1 − x) (1 − x ) (1 − x)(1 − x)3
x+2
=
.
(1 − x)(1 − x2 )(1 − x3 )
(36)
(−1)3 rOACc (1/x) = EOACc (x) + EADc (x) + EDc Ec′ (x)
1
1
1
+
+
=
2
2
2
2
2
(1 − x) (1 − x ) (1 − x )
(1 − x )(1 − x3 )
x2 + 2x + 3
=
.
(1 − x2 )2 (1 − x3 )
(37)
Next is OACc :
The last facet is OBc Cc :
(−1)3 rOBc Cc (1/x) = EOBc Cc (x) + EOEc′′ (x) + EBc Dc (x) + EDc Ec′′ (x) + EFc (x)
=
1
1
1
+
+
3
2
(1 − x)
(1 − x)(1 − x ) (1 − x)(1 − x2 )
+
=
(38)
2x2 + 1
1
+
2
2
(1 − x )
1 − x3
−2x5 + 4x2 + 5x + 5
.
(1 − x2 )2 (1 − x3 )
Finally, the edge OBc :
(−1)2 rOBc (1/x) = EOBc (x) =
33
1
.
(1 − x)2
(39)
Now Rmlc (x) results from (32), and then from (35) we see that
Lc (x) =
12x4
(
x2
Rmlc (x) =
(1 − x)2
)
79x15 + 190x14 + 260x13 + 250x12 + 211x11 + 179x10 + 181x9
+ 198x8 + 210x7 + 181x6 + 125x5 + 61x4 + 22x3 + 8x2 + 4x + 1
(1 − x4 )(1 − x5 )(1 − x3 )2 (1 − x2 )2
(40)
.
The constituents of Lc are extracted as described in Section 2.1, and here they are:

3 5 51 4 202 3 3769 2


 10 t − 8 t + 3 t − 10 t + c1 (t)t − c0 (t), if t is even;
(41)
Lc (t) =

3
51
202
7493

5
4
3
2
 t − t +
t −
t + c1 (t)t − c0 (t), if t is odd;
10
8
3
20
where c1 varies with period 6, given by

994,



986,
c1 (t) = 1909

,

2

 1925
,
2
if
if
if
if
t ≡ 0, 2
t≡4
t≡1
t ≡ 3, 5
and c0 , given by Table 7, varies with period 60.
34
(mod 6);
t
c0 (t)
t
c0 (t)
t
c0 (t)
t
c0 (t)
t
c0 (t)
0
948
12
948
24
36
76933
120
2780
3
35607
40
13292
15
18433
24
4452
5
18497
24
14332
15
32727
40
2572
3
93893
120
13
87301
120
14332
15
6891
8
11996
15
18433
24
4884
5
95941
120
2780
3
29271
40
2572
3
104261
120
25
29
41
4452
5
16769
24
14332
15
35607
40
2572
3
85253
120
48
1
4884
5
16769
24
13036
15
6891
8
13292
15
95621
120
30
948
42
948
54
31
85573
120
2780
3
32727
40
13292
15
20161
24
43
95941
120
14332
15
6315
8
11996
15
20161
24
55
4884
5
87301
120
2780
3
32151
40
2572
3
95621
120
4884
5
18497
24
13036
15
6315
8
13292
15
104261
120
2
3
4
5
6
7
8
9
10
11
14
15
16
17
18
19
20
21
22
23
26
27
28
32
33
34
35
37
38
39
40
44
45
46
47
49
50
51
52
53
56
57
58
59
Table 7: Constant terms of the constituents of Lc (t), counting all magilatin squares by upper
bound.
Thus the period of Lc turns out to be 60, just like that of Sc (not a surprise). However,
again as with Sc , the even constant terms have half the period of the odd constant terms.
That means Lc (2t) has period equal to half the denominator of the corresponding inside-out
polytope (2P, H). We have no explanation for this.
The principal constituent, that for t ≡ 0 (mod 60), is
3 5 51 4 202 3 3769 2
t − t +
t −
t + 994t − 948.
10
8
3
10
The constant term for magilatin squares does not have the simple interpretation as a number
of linear orderings that it does for magic and semimagic squares, because the entries in the
square need not all be different. (Still, there is an interpretation as a number of partial
orderings of the nine cells; see Theorem 4.1 in our general magic and magilatin paper [6],
and recall that an acyclic orientation of a graph can be represented by a partial ordering of
the vertices.)
We confirmed the formulas by generating all magilatin squares and comparing the count
with the coefficients of Lc (x) up to t = 91.
35
5.1.2
Symmetry types of magilatin squares, counted by upper bound
The number of symmetry types with strict bound t is lc (t). We count them via rmlc (w),
given by Equation (33); then
lc (t) =
t−1
X
(t − 1 − w) rmlc (w).
(42)
w=0
Equivalently, rmlc (w) counts the w1 -integral points in the interior and part of the boundary
of the inside-out polytope (Qc , Ic ) of Section 4.1.
We get rmlc (x) from (33); then from (42) we see that
lc (x) =
x4
(
x2
rmlc (x) =
(1 − x)2
9x15 + 20x14 + 23x13 + 16x12 + 10x11 + 13x10 + 27x9 + 43x8
+ 54x7 + 52x6 + 41x5 + 25x4 + 14x3 + 8x2 + 4x + 1
)
(1 − x2 )2 (1 − x3 )2 (1 − x4 )(1 − x5 )
The constituents of lc are:

1 5
3


t − t4 +

 240
64
lc (t) =



 1 t5 − 3 t4 +
240
64
(43)
.
97 3 2029 2
t −
t + ĉ1 (t)t − ĉ0 (t), if t is even;
216
720
97 3 4013 2
t −
t + ĉ1 (t)t − ĉ0 (t), if t is odd;
216
1440
where ĉ1 varies with period 6, given by
 17
,

2


 151 ,
18
ĉ1 (t) = 1163

,

144

 131
,
16
if
if
if
if
t ≡ 0, 2
t≡4
t≡1
t ≡ 3, 5
and ĉ0 , given by Table 8, varies with period 60.
36
(mod 6),
t
ĉ0 (t)
0
9
12
1
49213
8640
235
27
2823
320
1144
135
12313
1728
41
5
12953
1728
1229
135
2503
320
218
27
63293
8640
13
2
3
4
5
6
7
8
9
10
11
t
14
15
16
17
18
19
20
21
22
23
ĉ0 (t)
t
ĉ0 (t)
t
ĉ0 (t)
t
ĉ0 (t)
9
24
36
25
29
41
41
5
11225
1728
1229
135
2823
320
218
27
54653
8640
48
59581
8640
1229
135
539
64
982
135
12313
1728
47
5
68221
8640
235
27
2119
320
218
27
73661
8640
47
5
11225
1728
1067
135
539
64
1144
135
65021
8640
30
9
42
9
54
31
57853
8640
235
27
2503
320
1144
135
14041
1728
43
68221
8640
1229
135
475
64
982
135
14041
1728
55
47
5
59581
8640
235
27
2439
320
218
27
65021
8640
47
5
12953
1728
1067
135
475
64
1144
135
73661
8640
26
27
28
32
33
34
35
37
38
39
40
44
45
46
47
49
50
51
52
53
56
57
58
59
Table 8: Constant terms of the constituents of lc (t), counting symmetry types of magilatin
squares by upper bound.
Thus the period of lc turns out to be 60. As with Sc and Lc , the period of the even
constant terms is half that of the odd constant terms.
The principal constituent of lc is
3
97 3 2029 2 17
1 5
t − t4 +
t −
t + t − 9.
240
64
216
720
2
5.1.3
Some real numbers
For the first several nonzero values of the numbers of magilatin squares and of symmetry
types, consult this table:
t
Lc (t)
lc (t)
Rmlc (t)
rmlc (t)
4 5
6
7
8
9
10
11
12
13
14
15
12 48 120 384 1068 2472 4896 9072 15516 25608 40296 61608
1 4 10 24
53
106 191 328
528
822
1230 1794
12 24 36 192 420 720 1020 1752 2268 3648 4596 6624
1 2
3
8
15
24
32
52
63
94
114
156
The third line contains the number of symmetry classes of 3 × 3 magilatin squares, counted
by upper bound. The main numbers, Lc (t) and lc (t), are sequences A173548 and A173729
in the OEIS [15]. The reduced numbers, Rmlc (t) and rmlc (t), are sequences A174018 and
A174019. In contrast to the semimagic case, the number of squares is not a simple multiple
of the number of symmetry types.
37
5.2
Magilatin squares: Affine count (by magic sum)
The last example is 3 × 3 magilatin squares, counted affinely. Let La (t) be the number of
3 × 3 magilatin squares with magic sum t > 0.
The weak quasipolynomial is the same as in affine semimagic.
5.2.1
Magilatin squares by magic sum
We compute La (t), the number of squares with magic sum t, via Rmla (s), the number of
reduced squares with magic sum s. The formula is
X
Rmla (s)
if t > 0.
(44)
La (t) =
0<s≤t−3
s≡t (mod 3)
Equivalently, Rmla (s) counts 1s -integral points in the interior and part of the boundary of the
inside-out polytope (Qa , Ia ) of Section 4.2, each weighted by 72/|GS |.
Now we calculate (by LattE) the closed Ehrhart generating function for each necessary
face. First, OABa :
(−1)3 rOABa (1/x) = EOABa (x) + EOEa (x)
1
1
+
=
2
2
(1 − x)(1 − x )
(1 − x)(1 − x4 )
2
.
=
(1 − x)(1 − x2 )(1 − x4 )
(45)
Next is OACa :
(−1)3 rOACa (1/x) = EOACa (x) + EADa (x) + EDa Ea′ (x)
1
1
1
=
+
+
2
2
2
3
3
(1 − x)(1 − x )
(1 − x )(1 − x ) (1 − x )(1 − x4 )
x3 + x2 + x + 3
=
.
(1 − x2 )(1 − x3 )(1 − x4 )
(46)
The last facet is OBa Ca :
(−1)3 rOBa Ca (1/x) = EOBa Ca (x) + EOEa′′ (x) + EBa Da (x)
+ EDa Ea′′ (x) + EFa (x)
1
1
1
+
+
=
2
2
4
2
(1 − x)(1 − x )
(1 − x)(1 − x ) (1 − x )(1 − x3 )
1
1
+
+
3
4
(1 − x )(1 − x ) 1 − x5
(47)
x5 + 4x4 + 6x3 + 7x2 + 7x + 5
.
=
(1 − x5 )(1 − x3 )(1 − x4 )(1 + x)
Finally, the edge OBa :
(−1)2 rOBa (1/x) = EOBa (x) =
38
1
.
(1 − x)(1 − x2 )
(48)
Now we get Rmla (x) from (32); then by (44) we deduce that
x3
Rmla (x)
1 − x3


2
3
4
5
6
7
8
1
+
3x
+
7x
+
15x
+
33x
+
65x
+
128x
+
208x
+
316x






 + 434x9 + 566x10 + 676x11 + 784x12 + 852x13 + 911x14 + 936x15 
12x6


+ 967x16 + 967x17 + 1001x18 + 995x19 + 1000x20 + 955x21





22
23
24
25
26
27
28 
+ 893x + 752x + 624x + 456x + 322x + 174x + 79x
.
=
(1 + x)(1 + x + x2 )(1 + x2 )(1 − x3 )(1 − x5 )(1 − x6 )(1 − x7 )(1 − x8 )
La (x) =
(49)
The constituents of La are the following:
1
La (t) = t4 − 3t3 + a2 (t)t2 − a1 (t)t + a0 (t) − 72S7 (t),
8
where a2 varies with period 6, given by
 151
,

4


 135 ,
a2 (t) = 634

,

2

 71
,
2
if
if
if
if
t≡0
t ≡ 2, 4
t ≡ 1, 5
t≡3
the linear coefficient varies with period 12, given by

1296

, if t ≡ 0

5


1347

, if t ≡ 1, 5


10


831

,
if t ≡ 2, 10

5


 2097 , if t ≡ 3
10
a1 (t) = 876
 5 ,
if t ≡ 4, 8



1251


, if t ≡ 6

5


1257

 10 , if t ≡ 7, 11


 2187
, if t ≡ 9
10
(50)
(mod 6);
(mod 12);
the constant term a0 , given by Table 9, varies with period 120; and S7 is as in the affine
semimagic count.
39
t
a0 (t)
0
876
20
340
40
400
60
1
21
49
3283
40
3597
5
−941
40
1484
5
4887
8
821
5
2131
40
3948
5
5107
40
61
29
21843
40
1037
5
−461
40
4164
5
1367
8
881
5
17811
40
1388
5
5587
40
41
9
4243
40
1097
5
15219
40
1604
5
1463
8
3201
5
3091
40
1448
5
21267
40
10
265
30
705
50
11
−1037
40
4092
5
4819
40
809
5
4023
8
1676
5
5011
40
3273
5
787
40
31
1363
40
1772
5
19539
40
1109
5
311
8
3876
5
7411
40
593
5
18387
40
51
2
3
4
5
6
7
8
12
13
14
15
16
17
18
19
t a0 (t)
22
23
24
25
26
27
28
32
33
34
35
36
37
38
39
t a0 (t)
42
43
44
45
46
47
48
52
53
54
55
56
57
58
59
t a0 (t)
t
a0 (t)
840
80
376
100
364
81
89
20403
40
1217
5
−1901
40
3984
5
1655
8
701
5
19251
40
1568
5
4147
40
101
69
5683
40
917
5
16659
40
1784
5
1175
8
3381
5
1651
40
1268
5
22707
40
109
4723
40
3417
5
499
40
1664
5
4599
8
1001
5
691
40
3768
5
6547
40
241
70
229
90
741
110
205
16083
40
1712
5
3859
40
3309
5
791
8
1556
5
22131
40
893
5
−173
40
71
403
40
4272
5
3379
40
989
5
3735
8
1496
5
6451
40
3093
5
2227
40
91
−77
40
1592
5
20979
40
929
5
599
8
4056
5
5971
40
773
5
16947
40
111
17523
40
1892
5
2419
40
3489
5
503
8
1376
5
23571
40
713
5
1267
40
62
63
64
65
66
67
68
72
73
74
75
76
77
78
79
82
83
84
85
86
87
88
92
93
94
95
96
97
98
99
t a0 (t)
102
103
104
105
106
107
108
112
113
114
115
116
117
118
119
Table 9: Constant terms of the truncated constituents of La (t), counting magilatin squares
by magic sum.
The period of La turns out to be 840—the period of the constant terms, due to the
combination of a0 (t) and the constant term of S7 (t). This is equal to the denominator.
The principal constituent of La , that is, for t ≡ 0 (mod 840), is
1 4
151 2 9192
t − 3t3 +
t −
t + 948
8
4
35
(incorporating the −72S7 term). (As with the cubical magilatin count, there is an interpretation of the constant term 948 in terms of partial orderings; see Theorem 4.7 in our general
paper [6].)
40
We verified the results by comparing an actual count of magilatin squares with magic
sum t ≤ 100 to the coefficients in La .
5.2.2
Symmetry types of magilatin squares, counted by magic sum
We compute la (t), the number of symmetry types of squares with magic sum t, via rmla (s),
the number of reduced symmetry types with magic sum s. The formula is
X
rmla (s)
if t > 0.
(51)
la (t) =
0<s≤t−3
s≡t (mod 3)
Equivalently, rmla (s) counts 1s -integral points in the interior and part of the boundary of the
inside-out polytope(Qa , Ia ) of Section 4.2.
From (32) we get rmla (x) and then from (51) we see that
la (x) =
x3
rmla (x)
1 − x3


2
3
4
5
6
7
8
9
1
+
3x
+
7x
+
13x
+
23x
+
37x
+
60x
+
86x
+
118x
+
149x







 + 180x10 + 199x11 + 212x12 + 208x13 + 196x14 + 171x15
6
x


+ 145x16 + 115x17 + 96x18 + 79x19 + 72x20 + 67x21






22
23
24
25
26
27
28
+ 66x + 59x + 54x + 43x + 33x + 19x + 9x
.
=
(1 + x)(1 + x + x2 )(1 + x2 )(1 − x3 )(1 − x5 )(1 − x6 )(1 − x7 )(1 − x8 )
The constituents of la are:
la (t) =
1
1 4
t − t3 + â2 (t)t2 − â1 (t)t + â0 (t) − S7 (t),
576
48
where the quadratic term varies with period 6, given by
 25
,
if t ≡ 0


 96

25
, if t ≡ 1, 5
(mod 6);
â2 (t) = 144
59

,
if
t
≡
2,
4

288

 11
,
if t ≡ 3
48
the linear term varies with period 12, given by

31

,
if t ≡ 0

15


103

, if t ≡ 1, 5


120


133
 , if t ≡ 2, 10

120


 47 ,
if t ≡ 3
â1 (t) = 30
37
 30 ,
if t ≡ 4, 8



233


, if t ≡ 6

120


11

 15 ,
if t ≡ 7, 11


 203
, if t ≡ 9
120
41
(mod 12);
(52)
and â0 , given by Table 10, varies with period 120.
t
â0 (t)
0
8
20
1
2027
2880
553
360
979
320
229
90
847
576
221
40
1739
2880
104
45
1427
320
149
72
−1813
2880
73
10
2891
2880
301
360
279
64
128
45
2219
2880
233
40
−277
2880
21
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
t â0 (t)
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
47
18
1523
320
493
360
−949
2880
38
5
751
576
409
360
1171
320
193
90
3083
2880
49
8
587
2880
131
45
1299
320
601
360
−17
576
69
10
4619
2880
157
360
1267
320
t
â0 (t)
40
31
9
1067
2880
257
40
−1429
2880
199
90
343
64
349
360
779
2880
36
5
2603
2880
125
72
1043
320
247
90
1931
2880
229
40
463
576
113
45
1491
320
457
360
−1237
2880
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
t â0 (t)
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
15
2
3467
2880
373
360
1139
320
137
45
559
576
241
40
299
2880
163
90
1587
320
113
72
−373
2880
39
5
1451
2880
481
360
247
64
211
90
3659
2880
213
40
1163
2880
t
â0 (t)
80
28
9
1363
320
673
360
−2389
2880
71
10
1039
576
229
360
1331
320
119
45
1643
2880
53
8
−853
2880
217
90
1459
320
421
360
271
576
37
5
3179
2880
337
360
1107
320
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
t â0 (t)
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
53
18
2507
2880
237
40
11
2880
122
45
311
64
529
360
−661
2880
67
10
4043
2880
89
72
1203
320
146
45
491
2880
249
40
175
576
181
90
1651
320
277
360
203
2880
Table 10: Constant terms of the truncated constituents of la (t), the number of symmetry
types of magilatin squares with given magic sum.
The period of la is 840. The principal constituent, that for t ≡ 0 (mod 840), is
1
25
74
1 4
t − t3 + t2 − t + 9.
576
48
96
35
(This incorporates the −S7 term.)
42
5.2.3
Some numbers
The first several nonzero values are given in the table. The third line gives the number of
symmetry classes of squares.
t
La (t)
la (t)
Rmla (t)
rmla (t)
6 7 8 9 10 11 12 13
14
15
16
17
18
19
12 12 24 72 156 240 552 600 1020 1548 2004 2568 4008 4644
1 1 2 4
7
10 20 22
35
50
63
78
116 131
12 12 24 60 144 216 480 444 780 996 1404 1548 2640 3696
1 1 2 3
6
8
16 15
25
30
41
43
66
68
The sequences La (t) and la (t) are A173549 and A173730 in the OEIS [15]. The numbers
Rmla (t) of reduced magilatin squares and rmla (t) of normalized, reduced squares with largest
value t are sequences A174020 and A174021.
6
Observations and conjectures
A remarkable fact is that the period of every one of our strong Ehrhart quasipolynomials
equals the denominator, when it could be much smaller.
For some small values of t we calculated Ma (t) = EP◦ ◦ ,H(t) by hand, which is feasible because the problem is 2-dimensional. The process of counting lattice points in a diagram drew
our attention to some remarkable phenomena that apply to the semimagic and magilatin
problems as well. Let δ := dim s; let ck be the coefficient in the quasipolynomial EP◦ ◦ ,H(t)
w
and let cw
k be that in the Ehrhart quasipolynomial of P , and let pk , pk be their periods. We
w
observe that the variation in cδ−1 is exactly the same as that in cδ−1 , i.e.,
w
cδ−1 (t) − cδ−1 (t − 1) = cw
δ−1 (t) − cδ−1 (t − 1),
but that is not so for most lower coefficients, especially c0 . The reason is that adding each
new excluded hyperplane results in a constant deduction in degree δ − 1 (as we discussed at
Equation (4.9) in our first article [5]) but a more irregular one in lower terms. We observe
that pk increases—that is, there is longer-term variation in ck —as k decreases in every case.
Thus we propose some daring conjectures.
Conjecture 3. In an inside-out counting problem, let δ := dim P .
w
w
(a) pw
k | pk for 0 ≤ k ≤ δ. (We know that pδ−1 = pδ−1 because cδ−1 and cδ−1 have the same
variation.)
w
(b) If pj = pw
j for all j ≥ k, then the variation in ck is the same as that in ck .
(c) The period ratios increase by a multiplicative factor as k decreases:
pk pk−1
for 0 ≤ k ≤ δ.
pw
pw
k
k−1
We do not suggest pk | pk+1 because that is false in general in ordinary Ehrhart theory,
according to McAllister and Woods [13]. However, it might be true for the kinds of insideout polytopes that arise in cubical and affine counting.
43
7
Acknowledgments
The authors are grateful to the anonymous referee for several helpful suggestions.
References
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latte/
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2010 Mathematics Subject Classification: Primary 05B15; Secondary 05A15, 52B20, 52C35.
Keywords: magic square, semimagic square, magic graph, latin square, magilatin square,
lattice-point counting, rational convex polytope, arrangement of hyperplanes.
(Concerned with sequences A108235, A108236, A108576, A108577, A108578, A108579, A173546,
A173547, A173548, A173549, A173723, A173724, A173725, A173726, A173727, A173728,
A173729, A173730, A174018, A174019, A174020, A174021, A174256, and A174257.)
Received March 9 2010; revised version received June 1 2010. Published in Journal of Integer
Sequences, June 2 2010. Revised, June 8 2010.
Return to Journal of Integer Sequences home page.
45